Condensed Matter Physics, Nanoscience and Mesoscopic Physics

Summary

We have discussed the physical and mathematical aspects of one-dimensional quantum particles with1/r2 interactions. On the physics side, we began with the simplest two-body system, and culminated in the exact dynamics of a supersymmetric electronic model. In this way we have tried to draw a comprehensive picture of one-dimensional quantum systems, taking the canonical systems with 1/r2 interactions. On the mathematical side, the most important subjects are first the Jack polynomials, and second the Yangians. These two subjects were originally developed independently, but are actually closely interrelated. Our basic viewpoint is that the 1/r2 systems provide the most natural working model to synthesize these two subjects, and to sharpen the concepts further.

There are many topics, however, which we could not discuss in this book. This is firstly because of our insufficient understanding of the topics, and secondly because of our intention of making a reasonably sized book. We shall mention some of the omitted topics, hoping that they will be covered on another occasion.

Examples of other canonical systems, which have escaped our discussion in this book, include the Calogero–Sutherland model with harmonic confinement. The spectrum of the model is equidistant as in the simple harmonic oscillator. Because of this feature the model can be compared with the chiral conformal field theory (CFT) with right-going particles only. We refer to a recent review [88] for the systems. In relation to the CFT, there are active studies to investigate the quasi-particle structure for the general case of the internal symmetry.

Summary

In a quasisteady approximation, the mobile carriers in a forward-biased diode are assumed to follow the applied voltage instantaneously. This assumption is a good one for calculating the depletion-layer capacitance where the majority carriers are able to respond to the applied voltage virtually instantaneously. (See Section 2.1.4.8 for a discussion on majority-carrier response time.) However, the redistribution of minority carriers is through diffusion and recombination processes. These processes do not occur instantaneously, but on a time scale related to the minority-carrier lifetime or transit time. As a consequence, when a small signal is applied to a diode, the changes in the minority-carrier densities at different locations in the diode have different phases and cannot be lumped together and treated as a single entity. In this appendix, the diffusion capacitance is derived from a small-signal analysis of the current through a diode starting from the differential equations governing the transport of minority carriers (Shockley, 1949; Lindmayer and Wrigley 1965; Pritchard, 1967). The diffusion capacitance can also be obtained from a transmission-line analysis of a diode equivalent circuit (Bulucea, 1968).

Consider an n+–p diode with a time-dependent forward-bias voltage vBE(t) applied across it. The emitter is assumed to be wide, i.e., LpE << WE, where LpE is the hole diffusion length in the emitter and WE is the thickness of the emitter. The base is assumed to be narrow, i.e., LnB >> WB, where LnB is the electron diffusion length in the base and WB is the base width. (This diode represents the emitter–base diode of an n–p–n bipolar transistor.) We assume that vBE(t) consists of a small-signal voltage vbe(t) in series with a dc voltage VBE, i.e., vBE (t) = VBE + vbe(t). For simplicity, we assume parasitic resistances are negligible so that vBE (t) is the same as the emitter–base junction voltage. The current flows, including the displacement current, are shown schematically in Figure A6.1, where iE (t) is the emitter terminal current, iB (t) is the base terminal current, and CdBE,tot is the depletion-layer capacitance of the emitter–base junction. Overall charge neutrality or Kirchoff’s law requires that

iE(t) + iB(t) = 0.

Summary

Electron and hole generation and recombination processes play an important role in the operation of many silicon devices and in determining their current–voltage characteristics. Electron and hole generation and recombination can take place directly between the valence band and the conduction band, or indirectly via trap centers in the energy gap. Direct transitions involve energies larger than the bandgap energy Eg. Indirect transitions via trap centers involve energies smaller than Eg. As a result, indirect transitions are much more efficient than direct transitions. In this appendix, we derive the expressions for the generation and recombination rates due to indirect transitions via trap centers and examine the characteristics of generation and recombination currents.

Capture and Emission at a Trap Center

Consider a piece of silicon having in it a concentration of Nt trap centers per unit volume. For simplicity, we assume all of the trap centers to be identical and located at energy Et in the bandgap. Also, we assume that each trap center can exist in one of two charge states, namely neutral when it is not occupied by an electron and negatively charged when it is occupied by an electron. Each unoccupied center can capture an electron from the conduction band (electron capture). An electron in an occupied center can be emitted into the conduction band (electron emission). Similarly, each occupied center can capture a hole from the valence band (hole capture), and an unoccupied center can emit a hole into the valence band (hole emission). These four capture and emission processes are illustrated in Fig. A5.1. Note that in hole capture, the center turns from a negatively charged (occupied) state into a neutral (unoccupied) state. Hole capture from the valence band is equivalent to electron emission into the valence band. In hole emission, the center turns from a neutral (unoccupied) state into a negatively charged (occupied) state. Hole emission is equivalent to electron capture from the valence band.

Summary

The performance of a CMOS VLSI chip is measured by its integration density, switching speed, and power dissipation. CMOS circuits have the unique characteristic of practically zero standby power, which enables higher integration levels and makes them the technology of choice for most VLSI applications. This chapter examines the various factors that determine the switching speed of basic CMOS circuit elements.

Basic CMOS Circuit Elements

In a modern CMOS VLSI chip, the most important function components are CMOS static gates. In gate array circuits, CMOS static gates are used almost exclusively. In microprocessors and supporting circuits of memory chips, most of the control interface logic is implemented using CMOS static gates. Static logic gates are the most widely used CMOS circuit because of their simplicity and noise immunity. This section describes basic static CMOS circuit elements and their switching characteristics.

Circuit symbols for nMOSFETs and pMOSFETs are defined in Fig. 5.1. A MOSFET is a four-terminal device, although usually only three are shown. Unless specified, the body (p-substrate) terminal of an nMOSFET is connected to the ground (lowest voltage), while the body terminal (n-well) of a pMOSFET is connected to the power supply Vdd (highest voltage).

Summary

In this chapter, we give an example of how the Yangian theory works in practice in the calculation of physical quantities. Using the Yangian representation theory, calculation of the dynamical correlation function of the Sutherland model with glK symmetry can be performed in the same way as (a modified version of) the single-component Sutherland model [189]. Namely, Uglov [189] showed that the spin and the spatial momentum can be unified as a fictitious momentum. This spin–momentum unification is particularly powerful for exact derivation of spin correlation functions [189, 197]. Further development for the SU(K) chain has also been achieved [198, 199]. Uglov's theory is outlined in this chapter.

In Section 10.1 we introduce as a prelude the symmetric Macdonald polynomials, which include the symmetric Jack polynomials as a special case. Then in Section 10.2, Uglov symmetric polynomials are introduced as another limit of symmetric Macdonald polynomials. We explain in Section 10.3 an isomorphism between the Fock space of the Sutherland model with SU(2) internal symmetry and the space of Laurent symmetric polynomials. The isomorphism by which the Yangian Gelfand–Zetlin basis is mapped onto the Uglov polynomials preserves the inner product. By this isomorphism, density and spin-density operators in the U(2) model find their correspondence in the single-component model. In this way, the calculations of dynamical density and spin-density correlation functions in the U(2) Sutherland model reduce to those of a modified version of the single-component Sutherland model.

Macdonald symmetric polynomials

We have seen in Section 2.5 that the Jack symmetric polynomials can be defined as the homogeneous symmetric polynomials of z = (z1, …, zN) satisfying conditions of triangularity (2.177) and orthogonality (2.196).

Summary

A more or less generic CMOS process ﬂow is described below. It features shallow trench isolation (STI) (Davari et al., 1988, 1989), dual n+/p+ polysilicon gates (Wong et al., 1988; Sun et al., 1989), and self-aligned silicide (Ting et al., 1982). The front-end-of-the-line process consists of six or seven masking levels and is suitable for sub-0.5-µm generations of VLSI logic and SRAM technology.

Summary

This chapter treats quantum interference effects in disordered conductors. We have already discussed interference phenomena for the few-channels case in Section 1.6, and in Chapter 2 we have seen that the scattering approach becomes increasingly complicated for many transport channels. This is why we need special methods to treat many-channel diffusive conductors. Some of their properties can be understood if we replace the Hamiltonian or a scattering matrix by a random matrix. In Section 4.1 we discuss random matrices as mathematical objects and review the properties of their eigenvalues and eigenvectors. We then use random matrix theory to describe the properties of energy levels (Section 4.2) and transmission eigenvalues (Section 4.3).

To describe interference effects in a very broad class of conductors, we develop in Section 4.4 the methods to handle interference corrections for a circuit theory, which allows us to take account of all nanostructure details. We evaluate universal conductance fluctuations and the weak localization correction to conductance.

We also find that, in some situations, electrons are localized – confined to small regions of space. Section 4.5 discusses under which conditions this strong localization occurs and briefly outlines transport properties associated with this regime.

Random matrices

In this section, we describe disordered and chaotic systems. Disordered systems contain some defects that scatter electrons; chaotic ones do not have any defects, but scattering at the boundaries induces a very different motion for particles with very close energies. In these systems, the positions of energy levels and transmission eigenvalues are random and vary from sample to sample. We have already encountered such a situation when discussing the transport properties of diffusive conductors in Chapter 2.

Summary

This survival kit is intended to provide basic knowledge of superconductivity necessary for understanding the material of the book. We recommend Refs. [41], [56], and [183] to the reader who wishes to acquire a deeper understanding of superconductivity concepts.

Basic facts

Below a certain temperature Tc (the superconducting transition temperature), the electrical resistance of some metals vanishes. In particular, the most commonly used superconductors are aluminum and niobium; alkali, noble, and magnetic metals never become superconducting. The highest transition temperature found among pure metals is about 9K for niobium; among “usual” superconducting compounds it is 39K for magnesium diboride. There are compounds with even higher transition temperatures, of over 100 K, known as high-temperature superconductors. They possess unusual symmetries, which lead to very uncommon physical properties. We do not consider them in this book.
Superconductors are ideal diamagnets: weak magnetic fields do not penetrate the bulk of superconductors (Meissner effect). A high magnetic field destroys superconductivity. This critical field, Hc, can vary from 1G (approximately 10-4 T) for tungsten to 1980G for niobium.
There is a narrow layer at the boundary of the superconductor where an external magnetic field decreases exponentially to zero value in the bulk. The characteristic length of this decay δp, known as the penetration depth, is temperature-dependent and diverges at the transition temperature proportionally to (Tc - T)-12;.
The transition to the superconducting state is a second-order phase transition in zero magnetic field. It may become a first-order phase transition in finite magnetic fields.
[…]

Summary

In Chapter 3, we discussed charging effects in nanostructures: the most important manifestation of electron–electron interaction in quantum transport. In this chapter, we concentrate on another aspect of interactions which we have so far mentioned very briefly. It concerns interaction with slow modes. In most cases these slow modes are electromagnetic excitations in the nanostructure and nearby circuit that form an electromagnetic environment of the nanostructure. The effect of this interaction is threefold. First, it may affect and alter transport properties of the nanostructure. Secondly, it provides energy relaxation: transporting electrons and qubits may exchange their energy with the electromagnetic environment. Thirdly, the environment provides decoherence, inducing time-dependent phase shifts to wave functions of propagating electrons and qubits, thereby destroying the quantum coherence of corresponding states.

The physics discussed in this chapter is sometimes involved and various. It requires effort to see a “common denominator” in all effects mentioned. We choose to present material starting from the ideas of dissipative quantum mechanics: a branch of quantum mechanics developed in the 1970s and 1980s. For several concrete phenomena this presentation manner deviates from that commonly accepted in the literature. Although this may be inconvenient for the reader, we did this for the sake of the “big picture,” which allows us to see links and analogies between formally different phenomena.

The structure of the chapter is as follows. Sections 6.1 and 6.2 are introductory. In Section 6.1, we discuss electromagnetic excitations in linear circuits and the way to treat them quantum-mechanically. In Section 6.2 we review general ideas of dissipative quantum mechanics, which are not specific for quantum transport: the orthogonality catastrophe, shake-up, classification of environments.

Summary

It is difficult nowadays to graduate from a department of natural sciences and not hear anything about quantum computing, most likely about the fascinating prospects of it. Quantum computing by its origin is a rather abstract discipline, a branch of math or information science. It has emerged from a persistent search for more efficient ways to process information when quantum mechanics made it to the scope of the search. Being an abstract discipline, quantum computing approximates a physical quantum system with a number of axioms and explores the consequences of these axioms, precisely as conventional math has been. These activities begun in the 1970s, and for a long time it was not obvious why quantum calculational schemes have to be any better than common computer algorithms. This extensive work was rewarded with a break-through in 1994, when Peter Shor discovered a remarkable quantum algorithm for the factorization of large numbers into prime ones. This sounds quite abstract, but many public key cryptosystems will become obsolete if Shor's algorithm is ever implemented in a practical quantum computer. Modern communication security is based on the fact that the factorization is a tough problem for a classical computer: it takes too long to crack a code. The proposed algorithm speeds up the factorization enormously. The discovery by Shor has brought quantum computing to the scientific and even public attention.

This progress has motivated a massive research effort towards the manipulation of individual quantum systems, the practical realization of quantum computing schemes being one of the most attractive goals. This chapter describes this goal, the quantum transport systems suitable for it, and the achievements made so far.

Condensed Matter Physics, Nanoscience and Mesoscopic Physics

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Afterword

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Appendix 6 - Diffusion Capacitance of a p–n Diode

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Appendix 3 - Einstein Relations

Appendix 5 - Generation and Recombination Processes and Space-Charge-Region Current

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5 - CMOS Performance Factors

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10 - Uglov's theory

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Contents

Index

10 - Systematic derivation of the coupled-cluster equations

Appendix 1 - CMOS Process Flow

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1 - Scattering

Appendix C - Unit conversion

Appendix A - Survival kit for advanced quantum mechanics

4 - Randomness and interference

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Index

Appendix B - Survival kit for superconductivity

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Frontmatter

6 - Interaction, relaxation, and decoherence

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Contents

5 - Qubits and quantum dots

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